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19
Smoothing Spline ANOVA for Exponential Families, with Application to the Wisconsin Epidemiological Study of Diabetic Retinopathy
 ANN. STATIST
, 1995
"... Let y i ; i = 1; \Delta \Delta \Delta ; n be independent observations with the density of y i of the form h(y i ; f i ) = exp[y i f i \Gammab(f i )+c(y i )], where b and c are given functions and b is twice continuously differentiable and bounded away from 0. Let f i = f(t(i)), where t = (t 1 ; \De ..."
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Cited by 83 (44 self)
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Let y i ; i = 1; \Delta \Delta \Delta ; n be independent observations with the density of y i of the form h(y i ; f i ) = exp[y i f i \Gammab(f i )+c(y i )], where b and c are given functions and b is twice continuously differentiable and bounded away from 0. Let f i = f(t(i)), where t = (t 1 ; \Delta \Delta \Delta ; t d ) 2 T (1)\Omega \Delta \Delta \Delta\Omega T (d) = T , the T (ff) are measureable spaces of rather general form, and f is an unknown function on T with some assumed `smoothness' properties. Given fy i ; t(i); i = 1; \Delta \Delta \Delta ; ng, it is desired to estimate f(t) for t in some region of interest contained in T . We develop the fitting of smoothing spline ANOVA models to this data of the form f(t) = C + P ff f ff (t ff ) + P ff!fi f fffi (t ff ; t fi ) + \Delta \Delta \Delta. The components of the decomposition satisfy side conditions which generalize the usual side conditions for parametric ANOVA. The estimate of f is obtained as the minimizer...
Finding Chaos in Noisy Systems
, 1991
"... In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is comp ..."
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Cited by 50 (1 self)
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In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (18201900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.
Bayesian Smoothing and Regression Splines for Measurement Error Problems
 Journal of the American Statistical Association
, 2001
"... In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely dicult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this paper we describe Bayes ..."
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Cited by 21 (7 self)
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In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely dicult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this paper we describe Bayesian approaches to modeling a exible regression function when the predictor variable is measured with error. The regression function is modeled with smoothing splines and regression P{splines. Two methods are described for exploration of the posterior. The rst is called iterative conditional modes (ICM) and is only partially Bayesian. ICM uses a componentwise maximization routine to nd the mode of the posterior. It also serves to create starting values for the second method, which is fully Bayesian and uses Markov chain Monte Carlo techniques to generate observations from the joint posterior distribution. Using the MCMC approach has the advantage that interval estimates that directly model and adjust for the measurement error are easily calculated. We provide simulations with several nonlinear regression functions and provide an illustrative example. Our simulations indicate that the frequentist mean squared error properties of the fully Bayesian method are better than those of ICM and also of previously proposed frequentist methods, at least in the examples we have studied. KEY WORDS: Bayesian methods; Eciency; Errors in variables; Functional method; Generalized linear models; Kernel regression; Measurement error; Nonparametric regression; P{splines; Regression Splines; SIMEX; Smoothing Splines; Structural modeling. Short title. Nonparametric Regression with Measurement Error Author Aliations Scott M. Berry (Email: scott@berryconsultants.com) is Statistical Scientist,...
Bootstrap Confidence Intervals for Smoothing Splines and their Comparison to Bayesian `Confidence Intervals'
 J. Statist. Comput. Simulation
, 1994
"... We construct bootstrap confidence intervals for smoothing spline and smoothing spline ANOVA estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from exponential families. Several variations of bootstrap confidence intervals are considered and compared. ..."
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Cited by 16 (6 self)
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We construct bootstrap confidence intervals for smoothing spline and smoothing spline ANOVA estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from exponential families. Several variations of bootstrap confidence intervals are considered and compared. We find that the commonly used bootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property. Keywords: BAYESIAN CONFIDENCE INTERVALS, BOOTSTRAP CONFIDENCE INTERVALS, PENALIZED LOG LIKELIHOOD ESTIMATES, SMOOTHING SPLINES, SMOOTHING SPLINE ANOVA'S. 1 Introduction Smoothing splines and smoothing spline ANOVAs (SS ANOVAs) have been used successfully in a bro...
Behavior near zero of the distribution of GCV smoothing parameter estimates for splines
 Statistics and Probability Letters
, 1993
"... It has been noticed by several authors that there is a small but nonzero probability that the GCV estimate 2 of the smoothing parameter in spline and related smoothing problems will he extremely small, leading to gross undersmoothing. We obtain an upper bound to the probability that the GCV functio ..."
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Cited by 8 (6 self)
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It has been noticed by several authors that there is a small but nonzero probability that the GCV estimate 2 of the smoothing parameter in spline and related smoothing problems will he extremely small, leading to gross undersmoothing. We obtain an upper bound to the probability that the GCV function, whose minimizer provides,~, has a (possibly local) minimum at 0. This upper bound goes to 0 exponentially fast as the sample size gets large. For the mediumto smallsample case we study this probability both by Monte Carlo evaluation of a formula for the exact probability that the GCV function has a minimum at 0 as well as by replicated calculations of ~..
Confidence Intervals for Nonparametric Curve Estimates Based on Local Smoothing
 J. Am. Stat. Assoc
, 1998
"... Numerous nonparametric regression methods exist which yield consistent estimators of function curves. Often one is also interested in constructing confidence intervals for the unknown function. Pointwise confidence intervals are available using globally crossvalidated smoothing spline (GCV) estim ..."
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Cited by 8 (0 self)
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Numerous nonparametric regression methods exist which yield consistent estimators of function curves. Often one is also interested in constructing confidence intervals for the unknown function. Pointwise confidence intervals are available using globally crossvalidated smoothing spline (GCV) estimation. When the function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 \Gamma ff on average but because bias in nonparametric estimation is not uniform, they do not hold the desired level uniformly at all design points. To deal with this problem, a new smoothing spline estimator is developed which uses a local crossvalidation (LCV) criterion to determine a separate smoothing parameter for each design point. The local smoothing parameters are then used to compute the point estimators of the regression curve and the corresponding pointwise confidence intervals. Incorporation of local information th...
Quantitative Study of Smoothing SplineANOVA Based Fingerprint Methods for Attribution of Global Warming
, 1999
"... A fingerprintbased method for climate change detection and attribution with some novel features is proposed. The method is based on a functional ANOVA (ANalysis Of VAriance) decomposition of a time and space signal, further decomposed into global timetrend and timetrend anomaly as a function of s ..."
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Cited by 3 (1 self)
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A fingerprintbased method for climate change detection and attribution with some novel features is proposed. The method is based on a functional ANOVA (ANalysis Of VAriance) decomposition of a time and space signal, further decomposed into global timetrend and timetrend anomaly as a function of space. The method estimates the signal as a component of forced minus background climate model output, and then uses a partial spline model to estimate and test for the existence of signal in historical data. The method is based on the classical detection of signal in noise, however there are several features apparently novel to the fingerprint literature, in particular, the analysis takes place directly in observation space, anomalies are tted directly and there is possibility for estimating certain parameters of covariance models for the historical data as part of the analysis. Simulation studies using climate model runs from GFDL and NCAR and historical data for NH Winter average...
Backfitting in smoothing spline ANOVA, with application to historical global temperature data
, 1996
"... In the attempt to estimate the temperature history of the earth using the surface observations, various biases can exist. An important source of bias is the incompleteness of sampling over both time and space. There have been a few methods proposed to deal with this problem. Although they can correc ..."
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Cited by 3 (2 self)
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In the attempt to estimate the temperature history of the earth using the surface observations, various biases can exist. An important source of bias is the incompleteness of sampling over both time and space. There have been a few methods proposed to deal with this problem. Although they can correct some biases resulting from incomplete sampling, they have ignored some other significant biases. In this dissertation, a smoothing spline ANOVA approach which is a multivariate function estimation method is proposed to deal simultaneously with various biases resulting from incomplete sampling. Besides that, an advantage of this method is that we can get various components of the estimated temperature history with a limited amount of information stored. This method can also be used for detecting erroneous observations in the data base. The method is illustrated through an example of modeling winter surface air temperature as a function of year and location. Extension to more complicated mod...
Penalized Multivariate Logistic Regression With A Large Data Set
, 1999
"... We combine a smoothing spline ANOVA model and a loglinear model to build a partly exible model for multivariate Bernoulli data. The joint distribution conditioning on the predictor variables is estimated. The conditional log odds ratio is used to measure the association between outcome variables. A ..."
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Cited by 2 (1 self)
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We combine a smoothing spline ANOVA model and a loglinear model to build a partly exible model for multivariate Bernoulli data. The joint distribution conditioning on the predictor variables is estimated. The conditional log odds ratio is used to measure the association between outcome variables. A numerical scheme based on the block onestep SORNewtonRalphson algorithm is proposed to obtain an approximate solution for the variational problem. It is proved for a special case that the approximate solution can achieve the same statistical convergence rate as the exact solution, but is much more computing ecient. We extend GACV (Generalized Approximate Cross Validation) to the case of multivariate Bernoulli responses. Its randomized version is fast and stable to compute. Simulation studies show that it is an excellent computational proxy for the CKL (Comparative KullbackLeibler) distance. It is used to adaptively select smoothing parameters in each block onestep SOR iteration. Approximate Bayesian condence intervals are obtained for the exible estimates of the conditional logit functions. Simulation studies are conducted to check the performance of the proposed method. Finally, the model is applied to twoeye observational data from the Beaver Dam Eye Study to examine the association of pigmentary abnormalities and various covariates. ii Acknowledgements I would like to express my deepest gratitude to my advisor, Professor Grace Wahba. She initiated the research described in this dissertation and her dedication to statistics has been a tremendous inspiration to me. During the course of this study we had many fruitful discussions and she provided me numerous insightful suggestions. I shall always appreciate her guidance which led me into the wonderful world of smo...
1 Smoothing splines as locally weighted averages
, 1989
"... A smoothing spline is a nonparametric estimate that is defined as the solution to a minimization problem. Because of the form of this definition it is most convenient to represent a spline estimate relative to an orthonormal set of basis functions. One problem with this representation is that it obs ..."
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A smoothing spline is a nonparametric estimate that is defined as the solution to a minimization problem. Because of the form of this definition it is most convenient to represent a spline estimate relative to an orthonormal set of basis functions. One problem with this representation is that it obscures the fact that a smoothing spline, like most other nonparametric estimates, is a local, weighted average of the observed data. This property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates. Although equivalent kernels have been identified for a sdi.oothing spline these functions are either not accurate enough for asymptotic approximations or are restricted to equally spaced points. Rather than improve these approximations, this work concentrates on bounding the size of the kernel function. It is shown that the absolute value of the equivalent kernel decreases exponentially away from its center. This bound is used to derive the asymptotic form for the pointwise bias of a first order smoothing spline estimate. The pointwise bias has the usual form that would be expected for a second order kernel estimate with a variable bandwidth. Another potential application of this bound is in establishing the consistency of a spline estimate uniformly with respect to the smoothing parameter. Such results are important for studying databased methods for selecting the smoothing parameter.